YAKOV ILICH FRENKEL

A. I. Anselm

On the night of January 23 of this year, the outstanding Soviet theoretical physicist, laureate of the Stalin Prize, corresponding member of the Academy of Sciences of the USSR, Yakov Ilich Frenkel, passed away.

Yakov Ilich was an example of a scientist with an exceptionally broad range of interests and talents. Many ideas and propositions first expressed by him have become a lasting possession of modern science.

Molecular theory of crystals and liquids, electronic theory of metals, the theory of the electronic properties of dielectrics and semiconductors, nuclear physics, geophysics—all owe many of their fundamental concepts to Ya. I.

We shall begin our survey of Ya. I.’s scientific activity with an account of his work on the molecular theory of crystals and liquids. It seems to us that in this field he obtained the most significant scientific results. The original naive conception, held by physicists, of crystals as systems completely ordered in space, and of liquids as completely disordered systems, inevitably had to be replaced by conceptions more closely corresponding to real phenomena. In 1923, Academician A. F. Ioffe expressed the idea of the possibility of the transition of an ion or atom from one of the sites of a crystal lattice, as a result of its thermal motion, into one of the neighboring interstitial positions. Ya. I., in a number of works, developed this idea, considering the thermodynamic equilibrium and the mechanism of motion of such a dissociated ion and the “hole” it leaves behind, and constructed a quantitative theory of the ionic electrical conductivity of crystals. Since then “Frenkel defects” have become an inseparable property of a real crystal. Soviet physicists widely used Ya. I.’s ideas in interpreting the electrical properties of ionic crystals and the electronic properties of semiconductors. In the well-known monograph by N. Mott and R. Gurney, Electronic Processes in Ionic Crystals, a large part of the second chapter is devoted to “Frenkel defects.”

In 1925 Ya. I. put forward the fruitful idea of an analogy between the liquid and solid (crystalline) states. In fact, the distances between neighboring atoms in liquids and in crystals are of the same order. This leads to approximately the same interaction between atoms in solids and liquids. Ya. I. emphasized that this circumstance must be reflected not only in the similarity of certain macroscopic properties of both aggregate states, but also in the similarity of their internal molecular structure. Subsequently, X-ray structural analysis undoubtedly indicated the existence of short-range order in liquids, i.e., the quasicrystalline character of the immediate surroundings of each molecule of a liquid. It is very important that

YAKOV ILYICH FRENKEL

Ya. I. pointed out not only the necessity of the existence of short-range order in liquids, bringing them closer to crystals, so to speak, in the structural-geometrical respect, but also the identical character of the thermal motion of atoms in a crystal and in a liquid. According to Ya. I., the molecules of a liquid, for the most part, oscillate about some equilibrium position, passing on the average after a time \(\tau=\tau_0 \exp(U/kT)\) into a new equilibrium position (\(\tau_0\) is the period of oscillation, \(T\) the absolute temperature, \(U\) the height of the potential barrier separating the two equilibrium positions). Frenkel’s formula, given above, is from the theorist’s point of view almost trivial, but it proved exceptionally fruitful in the interpretation and systematization of an enormous body of experimental material on diffusion, viscosity, and the electrical conductivity of liquids. More precise experiments, carried out over a broader temperature interval, showed the limited applicability of the formula, but its progressive role in understanding the phenomena occurring in liquids remains unquestionable. At the beginning of the 1930s Ya. I. published a number of interesting works devoted to orientational order-disorder in crystals and liquids. In these works the concept of orientational waves, i.e. waves of rotational oscillations of molecules in crystals, was introduced for the first time. The study of the simplest linear chain of oriented molecules with fixed centers led Ya. I. to the important conclusion that the limiting frequency of orientational waves is inversely proportional to the square root of the moment of inertia of the particles. To this same cycle of works belongs the theory of orientational melting in crystals. This phenomenon consists in the fact that in a crystal, at a certain definite temperature, long-range order in the orientations of the molecules disappears. If the crystal consists of polar molecules, then orientational melting manifests itself experimentally in the form of a jump in the electric polarization occurring at some definite temperature. The jump in heat capacity observed at definite temperatures in crystals of hydrogen chloride and methane can also in some cases be explained by orientational melting. Ya. I.’s idea of orientational waves received further development in the works of Soviet theorists and led to a complete theory of orientational-translational waves in molecular crystals, widely used for interpreting the spectra of combination scattering at low frequencies discovered by Soviet physicists.

Ya. I. made a substantial contribution to the theory of the kinetic elasticity of high polymers. It is known that the kinetic theory of elasticity of linear high polymers is based on the idea of the thermal motion (rotation) of the individual links of a macromolecule. Ya. I., together with S. E. Bresler, was the first to take into account the effect of hindered rotation of individual links of a macromolecule, bringing the theory itself closer to the conditions of the real interaction of atoms in high polymers. These works of Ya. I. received further fruitful development in the works of Soviet theorists on the rotational isomerism of polymer molecules.

A large group of Ya. I.’s works is devoted to the mechanical properties of crystals. Here it is first of all necessary to point to the works on the theory of plastic deformation, carried out jointly with T. A. Kontorova. In these works it was shown that in a regular crystal lattice a special type of motion of atoms is possible—the successive coordinated displacement of whole groups of atoms from one equilibrium position to another. Such a “deformation wave” propagates with velocity

\(v=c[1-W_0^2:W^2]^{\frac12}\),

where \(c\) is the velocity of sound in the crystal, \(W\) is the energy of the group of atoms participating in the displacement, and \(W_0\) is the minimum energy of shear formation. It is assumed that the basis of na

observed experimentally in the plastic deformation of crystals is the motion of particles of precisely this type. It is interesting to note that in 1949, eleven years after the appearance of the work of Ya. I. and T. A. Kontorova, the English physicists F. Frank and Eshelby published a number of papers in which they considered that same question from a macroscopic point of view, i.e., using the equations of the theory of elasticity. They obtained results exactly coinciding with the results obtained by Ya. I. and Kontorova.

In 1941 there appeared a paper by Ya. I. and T. A. Kontorova devoted to the statistical theory of the brittle strength of real crystals. Developing quantitatively an idea, expressed earlier by A. P. Aleksandrov and S. N. Zhurkov, that the influence of the scale factor on the brittle strength of crystals is of a purely statistical character, the authors obtained a formula relating the most probable value of the stress in a specimen at which it fractures to its volume. It should be noted that comparatively recently this formula received experimental confirmation.

The role of Ya. I. in the development of the modern electron theory of metals is very great. In 1916 A. F. Ioffe organized in Petrograd, at the Polytechnic Institute, a scientific seminar on physics. N. N. Semenov, P. I. Lukirsky, P. L. Kapitsa, Ya. I. Frenkel, Ya. G. Dorfman and others took part in this seminar. In discussing the question of the nature of the potential barrier that confines free electrons inside a metal, Ya. I., who at that time was 22 years old, came forward with his own point of view, which consisted in the assumption of the existence on the surface of a metal of a double electric layer. Although the work of Ya. I. was done before the appearance of quantum mechanics and quantum statistics, the idea of the existence of a double electric layer on the surface of a metal has not lost its significance up to the present time.

The prediction of Ya. I. concerning the heat capacity of the free electrons of a metal is highly significant. It is known that the classical electron theory of metals encountered insurmountable difficulties in the question of their heat capacity, which, contrary to classical ideas, turned out to be of the same magnitude as the heat capacity of dielectric crystals. This unfortunate mischance of free electrons in the thermal equilibrium of a metal was interpreted by Ya. I. (qualitatively) quite correctly, as the intra-atomic motion of them with possible transitions from one lattice site to a neighboring one. Ya. I. arrived at these conclusions on the basis of correct considerations regarding the continuous change in the intra-atomic motion of electrons upon condensation of metal atoms into a lattice.

Ya. I. was able to find a surprisingly simple language for the exposition of complex physical theories. With exceptional simplicity and visual clarity he obtained the properties of the Fermi gas at absolute zero temperature. Starting from the principle of minimum energy of the system and Pauli’s principle, he was able, using literally a single expression for the volume of the ball in momentum space, to obtain correct values for the zero energy, the zero pressure, etc. On the basis of the same simple considerations Ya. I. showed that twice the mean kinetic energy of the electrons in a metal is equal to their potential energy, taken with the opposite sign. This makes it possible, with considerable simplicity and clarity, to explain the nature of cohesion forces in metals and to determine the correct order of magnitude of the lattice constant.

Despite the considerable successes of the formal theory of ferromagnetism, the nature of the internal molecular field remained unexplained until 1928. The constant of the internal field of ferromagnets, measured experimentally, proved to be a thousand times larger than it should have been if the magnetic interaction of atoms were taken into account. Ya. I., together

with Ya. G. Dorfman, correctly discerned the cause of the enormous internal field in ferromagnetics in the quantum-mechanical exchange interaction of neighboring atoms.

A major role in the modern electronic theory of dielectrics and semiconductors is played by the concept of excitons, advanced by Ya. I. in 1931. Considering the absorption of light in dielectrics, Ya. I. arrived at the fruitful idea of two possible types of absorption—one connected with the transition of an electron from the filled band into the conduction band, and another in which the excited electron remains bound to its atom. However, the excitation, or “exciton,” does not remain bound to a definite lattice site but, like an electron of the conduction band or a hole of the filled band, can move freely throughout the crystal. This circumstance makes it possible to speak of exciton waves in the same way as we speak of modulated waves of conduction electrons. To an exciton, just as to an electron in the periodic field of a crystal, one may assign a definite quasimomentum. In the absorption of a quantum of light (a photon) and the formation of an exciton, the corresponding conservation laws must be fulfilled: of energy and of momentum. Since in the process under consideration a quantum of Debye thermal waves (a phonon) may at the same time take part, the structure of the absorption spectrum becomes greatly complicated (the multiplet structure of the exciton). According to Ya. I., the transformation of an exciton into thermal vibrations may occur in two ways. In the first case the exciton moves so rapidly that it does not have time to deform the lattice near the excited atom of the crystal; then the dissipation of the exciton excitation takes place through the emission of individual thermal phonons. In the second case the “adhering” exciton deforms, to one degree or another, the lattice in its immediate surroundings, which leads to the possibility of the simultaneous emission of a larger number of phonons. One may say without exaggeration that the foundations of the electronic theory of selective absorption of light in dielectrics were laid by Ya. I. In recent years, in connection with certain experiments on the external photoelectric effect, interest in Frenkel excitons has greatly increased, and a number of theoretical works have appeared developing Ya. I.’s theory. The interesting works of the Soviet theorist S. I. Pekar on the theory of polarons, carried out mainly in 1944–1949, were likewise based on Ya. I.’s idea of the possible localization of a free electron in an ideal crystal lattice as a result of its polarization by the electron.

In 1931 Ya. I., together with Academician A. F. Ioffe, proposed a theory of the rectifying action of the contact of a metal with a semiconductor. The theory was based on a consideration of the mechanism of tunnel transitions of electrons between the metal and the semiconductor. The theory led to the correct dependence of the current on the applied potential difference. The further development of the theory of solid rectifiers, carried out chiefly by the works of Soviet theorists, showed the limited applicability of the theory of Ya. I. and A. F. Ioffe. However, a careful study of the question, carried out mainly in recent years, shows that there is no unified theory of solid rectifiers and that the “tunnel theory” is applicable in a number of technically interesting cases.

The enormous scientific activity of Ya. I. and his lively interest in the most topical problems of science could not but leave him indifferent to questions of nuclear physics. But here too, as everywhere, he brings his own distinctive approach to the question. First, he is little interested in those questions of nuclear physics in which the application of a cumbersome mathematical apparatus is not always justified by the physical value of the results obtained. Second, he contributes to the solution of those physical questions of nuclear physics that interest him, thermodynamic

and the statistical methods of theory, who had worked extensively in the field of molecular physics. This problem was facilitated for him by the fact that atomic nuclei (at any rate nuclei of heavy elements) are treated as charged drops—a model for whose development he himself had done so much.

In 1936 Ya. I. puts forward a bold idea: he proposes treating the emission of nucleons and alpha-particles by excited nuclei as a process analogous to the evaporation of atoms and molecules by a heated crystal. As in the theory of evaporation, he expresses the probability of emission of a particle by an intermediate nucleus (evaporation) in terms of the probability of the reverse process (condensation). Using this simple physical picture, Ya. I. calculates the width of nuclear energy levels and obtains a dependence of the width on the energy of the intermediate and final nuclei. When in 1939 the fission of uranium nuclei was experimentally discovered, Ya. I., even before the publication of the detailed calculations of Bohr and Wheeler, considered the energy questions connected with the fission of a heavy nucleus. Comparing the change of the surface nuclear and Coulomb energies of the drop-nucleus during its fission, Ya. I. comes to the conclusion that spontaneous fission of the uranium nucleus is possible—a phenomenon that was subsequently confirmed by brilliant experiments of Soviet physicists. He also brings a certain clarity to the dynamical theory of nuclear fission, which is governed by electrocapillary oscillations caused by neutron capture.

Ya. I.’s main scientific interests were directed toward solving concrete, topical problems of physics. This fully corresponded to his character and temperament, to his need for constant contact with experimental physicists, engineers, and scientific workers of other specialities: meteorologists, biologists, etc. However, these same qualities of his character did not prevent him from returning to the most fundamental problems of theoretical physics. In 1926, two years before the appearance of Dirac’s relativistic equation, Ya. I. published a paper in which he investigated the electromagnetic field of a moving electron possessing a magnetic moment. Proceeding from the laws of classical electrodynamics, Ya. I. shows that a moving moment necessarily also creates an electric moment. The requirement of relativistic invariance leads to the result that the components of the magnetic moment $\mathbf{m}$ and of the electric moment $\mathbf{p}$ are components of a certain antisymmetric tensor of the second rank. On the basis of these considerations, Ya. I. showed that the electric moment

\[ \mathbf{p}=\left[\mathbf{m}\,\frac{\mathbf{v}}{c}\right], \]

where $\mathbf{v}$ is the velocity of the electron and $c$ is the speed of light. It should be noted that Frenkel’s formula also follows from the exact Dirac equation for the motion of the electron.

Ya. I.’s interest in questions of geophysics—the theory of atmospheric electricity and the problem of terrestrial magnetism—appeared already in his youth. In 1912, while still a gymnasium student, Ya. I. tried to construct a theory of atmospheric electricity and a theory of terrestrial magnetism based on the idea of demagnetizing action on the Earth. Ya. I. showed his experiments to the young professor A. F. Ioffe, who subjected them to fair criticism, but noted the talented youth and later drew him into work at the Physico-Technical Institute. In 1944–1946 Ya. I. again turned to these same problems and, as it seems to us, introduced a substantial contribution into the theory explaining the existence of an electric field in the Earth’s atmosphere.

Observations show that near the surface of the Earth there always exists an electric field. In the absence of clouds this field is directed vertically downward (which corresponds to a negatively charged surface of the Earth) and in order of magnitude is equal to $1\ \mathrm{V/cm}$. In the presence of clouds, po-

...whereas at the surface of the Earth it has the opposite direction and a considerably larger magnitude. Ya. I.’s theory explains these facts well, both qualitatively and quantitatively. According to Ya. I., clouds are gigantic generators maintaining the Earth’s electric field, which tends to dissipate because of the electrical conductivity of the atmosphere caused by the action of cosmic rays and the radioactivity of the soil. Using the well-known fact that water droplets immersed in a neutral ionized atmosphere become negatively charged, Ya. I. examines in detail the mechanism of the fall of an electronegative cloud in the Earth’s gravitational field. If one takes into account the presence in the volume of the cloud of a positively charged atmosphere, it becomes easy to show that the mechanism considered produces an electric polarization of the cloud, creating in the surrounding space a field whose properties agree with observations.

In our brief survey of Ya. I.’s scientific activity we have not touched upon many of his interesting works, but we hope that we have mentioned the majority of the investigations that have left a substantial mark on science. The total number of works published by Ya. I. is very large—it exceeds two hundred. It remains incomprehensible how, with such enormous creative activity, he still had the time and strength to work on the numerous monographs and textbooks on which he labored unceasingly all his life. Mechanics, electrodynamics, the theory of relativity, quantum mechanics, statistical physics, the theory of metals, the kinetic theory of liquids, and many other questions of physics received in Ya. I.’s books an interesting, perhaps not always indisputable, but original exposition rich in new ideas. Of particular interest is Ya. I.’s monograph Kinetic Theory of Liquids, published by the Academy of Sciences in 1945 and, as it were, summarizing the author’s twenty years of work in this direction. For this monograph Ya. I. was awarded the Stalin Prize, first degree, in 1947. There is no doubt that Ya. I.’s interesting and original textbooks contributed much to the development of theoretical physics in the Soviet Union. Some of Ya. I.’s textbooks were subjected to just criticism by the Soviet scientific community. Ya. I. was able to draw the proper conclusions from this criticism—the Theory of Metals, republished by him in 1950, was to a considerable extent freed of the shortcomings that had previously been inherent in it. Ya. I.’s death left unfinished the course of Wave Mechanics that he had been revising—a work which he was never destined to bring to completion.

Looking back on the path traversed by this man during his not very long, 58-year life, one is struck by the enormous and noble labor he accomplished. The range of Ya. I.’s scientific interests was immense, beginning with the ultratheoretical questions of the relativistic theory of the electron and ending with technical problems of the sparking of tram wires. He showed exceptional interest in related fields of physics—geophysics, biophysics, astrophysics—which he also enriched with many interesting ideas.

The scientific seminar led by Ya. I. from the beginning of the twenties was for a long time one of the most active and advanced scientific centers in theoretical physics in the Soviet Union. Many physicists-theorists began their work while being connected with Ya. I. to one degree or another, and if many of them subsequently became major scientists themselves and found their own paths in science, they owe their first steps in the scientific arena to Ya. I. It is not without reason that to the last days of his life Ya. I. was connected with his students not only in Leningrad, but also in Moscow, Kiev, Sverdlovsk, Lvov, Chernovtsy, and other cities of the Union. Ya. I. combined such great work in training scientific personnel with pedagogical activity

at the Leningrad Polytechnic Institute, where for thirty years he headed a department in the Faculty of Physics and Mechanics.

It might seem that such an enormous amount of work could be accomplished only by a narrow scholar of the study-bound type. However, those who knew Ya. I. personally know how little the notion of a “study-bound scholar” applied to him. Even before the war Ya. I. devoted much attention and assisted the development in our country of scientific and educational cinematography. During the war, in 1944–1945, Ya. I. worked on certain problems at the Institute of Theoretical Geophysics in Moscow. From 1945 to 1947 he worked at the Main Geophysical Observatory in Leningrad, while remaining a member of the Scientific Council on the Physics of Life; even on his last visit to Moscow, in December 1951, Ya. I., despite feeling unwell, continued to consult staff members of the All-Union Institute of Aviation Materials.

The Government highly valued Ya. I.’s work, awarding him the Order of the Red Banner of Labor in 1945 and conferring on him the Stalin Prize, first class, in 1947.

Ya. I.’s outward life was not rich in events. Ya. I. was born on February 10, 1894, in Rostov-on-Don, but lived almost his entire life in Petersburg—Leningrad. After finishing secondary school in Petersburg, he entered the Faculty of Physics and Mathematics of Petersburg University, from which he graduated in 1917. In the spring of 1917 he and his family moved to Yalta, where after the October Revolution he took an active part in the organization of Taurida University. During the period of the establishment of Soviet power in the Crimea he worked in the People’s Commissariat of Education. In 1920, during the temporary occupation of the Crimea by the Whites, Ya. I. was imprisoned for this work.

After the final victory of Soviet power, Ya. I. returned in 1921 to Petrograd, where, at the invitation of Academician A. F. Ioffe, he began to work at the Leningrad Physico-Technical Institute and in the Faculty of Physics and Mechanics of the Polytechnic Institute. Here, in Lesnoye, between the cheerful, bright building of the Physico-Technical Institute and the austere bulk of the Polytechnic, the whole life of this major scholar and remarkable person unfolded.

I think there is not a single person who, upon personal acquaintance with Ya. I., did not feel the full charm of his personality, his boundless goodwill toward those around him. He was absolutely alien to the pose of the scholar-mentor, clothed in the toga of infallibility. His stormy temperament as a scholar sometimes led him into errors, but he never refused to reconsider his points of view. He was not only a scholar in one of the most abstract fields of human knowledge, but also a great lover of music and painting.

Such is the image that stands before us of this major scholar, honest citizen, and charming person.